The surface a table draws

Where closed-shell atoms come to rest asked a question that any contour drawing raises. Every orbital picture here is drawn at a stated fraction of the density; the fraction is a choice; and the awkwardness of the choice would go away if some fraction turned out to be the one at which atoms actually stop. It computed the separation at which two closed-shell atoms come to rest and asked what fraction of their density lay inside half of it. For helium, neon and argon the answers were 99.75, 99.53 and 99.38 per cent — three atoms, one number to two figures, and a very tempting conclusion.

That conclusion has an obvious test it had not been put to. “A fluoride ion’s outermost density extends much further than a neutral fluorine’s and its tabulated radius is much larger, so the same calculation would say what fraction a charged closed shell stops at — and whether the near-constancy above survives a change of charge is a question with a definite answer.”

It does not survive. What replaces it is more interesting than the near-constancy would have been, because the way it fails says something about the table rather than about the atoms.

The screening is the ion’s and the nucleus is the atom’s #

A fluoride ion has ten electrons and nine protons. Slater’s rules screen an electron by the other electrons, so the screening is neon’s; the nucleus it is screened from is fluorine’s. The rules as usually applied take an atomic number and build the configuration from it, which is right for a neutral atom and quietly wrong for an ion — using the same number twice gives either the wrong nucleus or the wrong screening.

Separating them makes the isoelectronic series into arithmetic. Every ten-electron species has a 2p electron screened by 4.15 — seven others in its own group at 0.35 apiece, two in the shell below at 0.85 — so the effective charges are 3.85, 4.85, 5.85, 6.85, 7.85 and 8.85 for O²⁻, F⁻, Ne, Na⁺, Mg²⁺ and Al³⁺. One proton, one unit of effective charge, exactly, all the way along.

That is the model, and it is the same one the radial distributions across the table were drawn from. It is a crude one — a Slater-screened hydrogenic shell is not a Hartree–Fock ion — and the reason it is good enough here is that the question is about ratios of lengths within one series, where the same crudeness applies to every member.

What the tables say, and what the density says #

Two kinds of radius are quoted for these species and they come from two kinds of measurement.

A van der Waals radius is fitted to the distance between two neutral atoms that are not bonded to each other. Bondi’s value for neon is 1.54 Å and for argon 1.88 Å, and the fitting is to contacts in crystals of the elements themselves, so the two atoms are alike and the split is by symmetry rather than by convention.

An ionic radius is one term of a sum. What is measured is the distance between a cation and an anion; the two radii are apportioned so that the sums reproduce a great many such distances at once. Shannon’s six-coordinate values are the standard set, and they are excellent at what they are for. But the split between the two terms is a decision, and the decision is not a measurement of either ion.

The numbers are not close to constant. Down the ten-electron series: 97.38 per cent for O²⁻, 99.31 for F⁻, 99.98 for neon, 99.66 for Na⁺, 98.10 for Mg²⁺, 94.38 for Al³⁺. Down the eighteen-electron series: 91.17, 96.84, 99.40, 97.40, 91.22. Two of those are below the fraction at which an orbital picture here is drawn and the space outside it called negligible — and the tenth that is not drawn is exactly the part of the density that decides these numbers.

The peak is at the neutral, and that is the tell #

The shape of both curves is the same and it is not the shape a physical trend would have. An anion’s density is diffuse, so a radius that is too small will enclose too little; a cation’s is compact, so a radius that is too small will enclose almost everything. A single convention applied across a series would give a curve that rose or fell. What these do is rise to the neutral and fall away again, symmetrically enough to look designed.

It is designed, in the sense that the neutral’s number came from somewhere else. It is a van der Waals radius and the ions’ are crystal radii, and the two are fitted to different measurements. The discontinuity is at exactly the row where the tabulation changes, and there is nothing physical at that row at all — neon and Na⁺ have the same ten electrons in the same shells, and their densities differ only by the one unit of nuclear charge that the effective charges already carry.

So the first answer to the question is: the near-constancy does not survive, and one third of the reason is that the comparison mixes two tables.

The additivity test, which is the real one #

The other two thirds are worth more, because they can be checked against something.

Take the convention this collection actually uses. Give every ion the radius at which its own density encloses a stated fraction — ninety-nine per cent, say — and ask whether two such radii add up to the distance between the ions in a crystal. That is not a rhetorical question: an isovalue is a choice, and if some choice made radii additive it would be the choice to make.

Sodium fluoride comes out 6.33 per cent too short. Calcium sulfide comes out 37.27 per cent too long. The errors run one way for the singly charged pairs of small ions and the other way for the doubly charged pairs of large ones, which is a systematic failure rather than scatter.

Nor is the fault in the particular fraction chosen. Ask instead what fraction would reproduce each measured distance exactly, and the answer is a different number for every pair: 99.42 per cent for NaF, 98.38 for NaCl, 97.54 for MgO, 95.47 for CaO, 94.34 for MgS, 91.26 for CaS.

Eight points spanning eight percentage points is not a constant with noise on it. It is a trend, and it runs the way a chemist would expect if the answer were about polarisation rather than about size — the more highly charged and the more diffuse the pair, the further into each other’s density they sit. That is a real effect and it is not in this model at all, which is why the model can display the failure and cannot repair it.

What a contour is a boundary of #

Put the two halves together and the tempting conclusion becomes a statement about neutral atoms of similar size, which is a much smaller claim than it looked.

There is a way of seeing why. An isosurface at a stated enclosure is a statement about one atom’s density integrated over all space, and it knows nothing about what is outside it. A contact distance is a statement about two atoms’ energy, which is a balance between a repulsion that goes as the square of the overlap and an attraction that goes as an inverse sixth power. Those two happen to agree closely for three closed shells of comparable diffuseness because the balance falls at a separation where the density has already decayed to a similar small value — not because the surface is what the atoms are stopping at.

The three noble-gas numbers are the whole of the evidence on the other side, and they are worth holding beside that: 99.75, 99.53 and 99.38 per cent, agreeing to within four tenths of one. Reading that agreement as evidence that a surface at ninety-nine and a half per cent is where an atom ends requires the same number to come back for something that is not a neutral closed shell of comparable diffuseness. It does not come back. What comes back instead is a spread of five and a half percentage points, and a peak in the wrong place.

Charge the shell and the two statements come apart, because charging it changes the balance and the density by different amounts. The density contracts as the effective charge rises — the ninety-nine-per-cent radius of the ten-electron series runs 1.601, 1.271, 1.053, 0.900, 0.785, 0.696 Å from O²⁻ to Al³⁺, a factor of 2.3 — while the crystal separation is set by an electrostatic attraction that is not in the one-electron picture at all.

There is a second reading of the same fact, and it is the one an orbital’s size has always had here: a length quoted for a distribution is a statement about which measure was taken, and four measures of the same orbital are four different lengths.

Where the two conventions can be compared honestly #

There is one comparison in this that is clean, and it is worth separating out from the rest.

Within one series, at one tabulation, the fraction still varies enormously: 99.31 per cent for F⁻ against 94.38 for Al³⁺, both from Shannon, both six-coordinate. Al³⁺'s tabulated radius of 0.535 Å is well inside the shell it is drawn around — its own ninety-per-cent radius is 0.475 Å, so 0.535 Å is only just outside the surface enclosing nine tenths — while F⁻'s 1.33 Å sits at 99.3 per cent, comfortably outside.

Five percentage points of enclosure is a much larger difference in length than the percentages suggest, and the reason is the shape of the curve the fraction is read off.

That difference is not an artefact of mixing tables. It is Shannon’s apportionment saying that a small highly charged cation is further inside its own electron cloud than a large singly charged anion is — which is exactly the polarisation statement above, arrived at from the other side, and is a real thing about ionic crystals rather than a defect of the fitting.

The object under all of this is a surface at a stated fraction — for a 2s, a shell inside a shell, since the radial node puts one closed surface inside another — and it is worth being clear that the object is not in doubt. It is well defined, it is reproducible to as many figures as the integration is run to, and every level above was solved for. What the additivity test denies is not that the surface exists but that two of them stop against each other.

The table the test is run against cannot fail it #

One asymmetry in the additivity test deserves stating, because it makes the result stronger rather than weaker and it is easy to read the other way.

A tabulated ionic radius is not a measurement. What is measured is a separation — the distance between two nuclei in a crystal — and a radius is one half of a division of that distance between the two ions. The division is not determined by the measurement: any pair of numbers adding to the observed separation reproduces it exactly. What fixes the split is a convention, historically an assumed value for one ion, with every other radius on the scale then derived from measured separations relative to it.

So a table built that way reproduces the separations it was built from by construction. Finding that Shannon’s radii add up to the measured distances within five per cent is not evidence that ionic radii are additive; it is evidence that the fitting worked, on the same data.

The comparison in this section is therefore between a convention that was fitted to these eight separations and a convention that was not fitted to anything. One of them cannot fail and the other has no way to succeed by accident, and the interesting question is not which wins.

It is where the second one fails, and by how much, and in which direction — because those are the only quantities in the comparison that carry information. A fixed-enclosure radius misses in both directions rather than by a constant offset, which is the finding, and it is a finding about the density rather than about the tables. Had it missed by a constant, the fraction could simply have been rechosen.

What this cannot say #

These are not ions. A Slater-screened hydrogenic shell has the right number of electrons and the wrong shape: no self-consistency, no exchange, and a radial function whose tail decays as a single exponential where a real one does not. What the model is being used for is the ratio of a tabulated length to a computed length, within a series where the same defects apply to every member, and the numbers should be read at two figures rather than four.

Nothing here computes a lattice. The interionic distances are quoted from measurement. Computing one would need an electrostatic lattice sum, whose value depends on the order of adding, and a repulsion between closed shells; only the second is computed here. What holds a solid together is where that boundary was drawn and why.

The additivity failure is not a criticism of Shannon. His radii are additive by construction and they work; the point is that additivity was put in rather than discovered, and that the surfaces which would make it discovered do not exist. A table that reproduces ten thousand distances is a good table whatever its terms mean individually.

And the enclosed fractions depend on the screening rule. Clementi’s effective charges differ from Slater’s by up to a third for a 2p — neon’s are 4.4532 and 5.85 — and the enclosed fractions would move accordingly. The pattern across a series would not, because the screening is the same for every member of one, and the pattern is what the argument uses. The same caution applies to comparing one contour against another: what is being compared here is a ratio of two lengths, not two absolute sizes.

What the comparison requires #

The screening is constant along an isoelectronic series and the effective charge rises by exactly one per proton, to twelve decimal places, for both series. That is what separates an ion’s configuration from its nucleus, and a routine that built the configuration from the atomic number would fail it at once.

The enclosure integral agrees with the closed form for a 1s at three effective charges and three fractions, to two parts in ten thousand — because a numerical integration of a density is precisely the kind of thing that returns a plausible wrong number.

The tabulated radius encloses some of the density and not all of it, for every one of the eleven species, which is the loose bound that would catch an integral that had gone wrong by an order of magnitude.

The enclosed fraction spans more than five percentage points across each series, against three tenths of one for the neutral atoms — the result, stated as an inequality so that a near-constancy would refuse it.

And it peaks at the neutral in both series, which is the check that identifies the discontinuity as the tabulation rather than the density.

The refusal is additivity. Shannon’s radii must reproduce the eight measured separations within five per cent, and radii at a fixed enclosure must miss them in both directions by more than five — so the failure is a failure of the convention rather than a constant offset that could be absorbed into the fraction.

Still open: how far a surface moves, and one set of ionic radii #

The original question — where an atom’s surface is — has an answer, and it is a negative one: there is no fraction at which a contour is the boundary of an atom, because the thing two atoms’ separation is set by is not a property of either one’s density alone.

What that leaves is the direction the additivity failure points in. The fitted fraction falls monotonically with the charge on the pair and with the size of the anion, and both of those are what a polarisability scales with. So the natural next step is to stop asking where an ion’s surface is and ask how much its surface moves when there is another ion beside it — which is a computable quantity, because a field mixes an s function with a p one and the mixing is the whole of a polarisability in the one-electron picture.

That would turn the eight discrepancies above from a refusal into a measurement: a fitted fraction per pair is a number, and if it correlates with a computed polarisation then the failure of a fixed-enclosure convention has a cause rather than merely a size.

The nearer question is smaller and is about the tables. The peak at the neutral is an artefact of comparing Bondi against Shannon, and it could be removed by computing a van der Waals radius for each ion from the same closed-shell contact calculation used for the noble gases — which is possible for the isoelectronic series here, since a Slater-screened shell is a Slater-screened shell whatever its charge. What comes out would be a single self-consistent set of radii to compare against both tables at once, and the comparison would say which of the two the enclosure convention is nearer to.